A system that has three unperturbed states can be represented by the perturbed hamiltonian matrix $$ \left[\begin{array}{lll} E_{1} & 0 & a \\ 0 & E_{1} & b \\ a^{*} & b^{*} & E_{2} \end{array}\right] $$where \(E_{2}>E_{1}\). The quantities \(a\) and \(b\) are to be regarded as perturbations that are of the same order and are small compared with \(E_{2}-E_{1} .\) Use the second-order nondegenerate perturbation theory to calculate the perturbed eigenvalues (is this procedure correct?). Then diagonalize the matrix to find the exact eigenvalues. Finally, use the second- order degenerate perturbation theory. Compare the three results obtained.

In quantum mechanics, the Hamiltonian matrix, commonly denoted as \(H\), is a key component that describes the total energy of the system. It includes both the kinetic and potential energies. For a system with three unperturbed states, the Hamiltonian matrix can be divided into two parts: the unperturbed Hamiltonian \(H_0\) and the perturbation \(V\).

The unperturbed Hamiltonian matrix in our exercise is given by:
\[H_0 = \left[\begin{array}{ccc} E_1 & 0 & 0 \ 0 & E_1 & 0 \ 0 & 0 & E_2 \end{array}\right]\]

This essentially describes the energy levels without any disturbance.

The perturbation \(V\) slightly alters these energies:

\[V = \left[\begin{array}{ccc} 0 & 0 & a \ 0 & 0 & b \ a^* & b^* & 0 \end{array}\right]\]

Here, the quantities \(a\) and \(b\) represent the small perturbations applied to the system. Understanding how this affects the system's energy levels involves perturbation theory, which describes the impact of such perturbations.

Eigenvalues are vitally important in quantum mechanics because they represent possible outcomes of measurements of physical properties such as energy. They are obtained by solving the characteristic equation for the Hamiltonian matrix.

For the perturbed Hamiltonian matrix \(H = H_0 + V\) mentioned in the exercise, we must solve the equation:

\[\det(H - \lambda I) = 0\]

This allows us to find the eigenvalues of \(H\), where \(\lambda\) is an eigenvalue, and \(I\) is the identity matrix. The eigenvalues can be interpreted as energy levels of the system.

In non-degenerate perturbation theory, the perturbed eigenvalues \(E_i^{(2)}\) for state \(i\) are corrected to the second order as:
\[E_i^{(2)} = E_i + \left\langle i \right|V\left| i \right\rangle + \sum_{j eq i} \frac{\left|\left\langle j \right|V\left| i \right\rangle\right|^2}{E_i - E_j}\]

This accounts for the influence of the perturbation on the original energy levels.
By diagonalizing the perturbed Hamiltonian matrix, we can solve for the exact eigenvalues, providing a precise understanding of how the perturbation affects the system.

Comparing these results helps to evaluate the accuracy and significance of the perturbative approach.

Perturbation theory is used to find approximate solutions to systems where the Hamiltonian is slightly altered from a system with known solutions. There are two main types: non-degenerate and degenerate perturbation theory.

In non-degenerate perturbation theory, the original energy levels (eigenvalues) are distinct. This theory is simpler because we can treat each state independently.

The second-order non-degenerate perturbation correction to the energy is given by:
\[E_i^{(2)} = E_i + \left\langle i \right|V\left| i \right\rangle + \sum_{j eq i} \frac{\left|\left\langle j \right|V\left| i \right\rangle\right|^2}{E_i - E_j}\]

Degenerate perturbation theory deals with the case where two or more energy levels are the same (degenerate). Special treatment is needed because the perturbation can mix degenerate states.

The effective Hamiltonian for the degenerate subspace is:
\[V_{eff}^{(2)} = \left[\begin{array}{cc}\sum_{k} \frac{\left|\left\langle k \right|V\left|1 \right\rangle\right|^2}{E_1 - E_k} & \sum_{k} \frac{\left\langle 1 \right|V\left| k\right\rangle \left\langle k \right|V\left| 2 \right\rangle}{E_1 - E_k} \ \sum_{k} \frac{\left\langle 2 \right|V\left| k \right\rangle \left\langle k \right|V\left| 1 \right\rangle}{E_1 - E_k} & \sum_{k} \frac{\left|\left\langle k \right|V\left|2 \right\rangle\right|^2}{E_1 - E_k} \end{array}\right]\]
Understanding both types of perturbation theory helps to evaluate the effect of perturbations accurately whether the states are degenerate or not.